How can there be an inner product space when inner product yields a scalar?

Question

I thought the inner product yields a scalar in both real and complex cases. How can a space be made up of scalars? Taking two vectors $a$ and $b$, $a=(a_1,a_2)$ and $b=(b_1,b_2)$, the inner product is $$( a,b)=(a_1\overline{b_1}+a_2\overline{b_2})$$ That's a scalar. How can there be a space of scalars?

Answer 1

An inner product space is a linear space, equipped with an inner product on it. So, it's a pair $(X,\langle\cdot,\cdot\rangle)$, where $X$ is the linear space and $\langle\cdot,\cdot\rangle$ the inner product on $X$.